Loading...

PPT – On the Lagrangian theory of cosmological density perturbations PowerPoint presentation | free to download - id: 125477-OTEzM

The Adobe Flash plugin is needed to view this content

On the Lagrangian theory of cosmological density

perturbations

V. Strokov Astro Space Center of the P.N.

Lebedev Physics Institute Moscow, Russia

Isolo di San Servolo, Venice Aug 30, 2007

Outline

- Cosmological model
- Scalar perturbations
- Hydrodynamical approach
- Field approach
- Isocurvature perturbations
- Conclusions

Cosmological model

Background Friedmann-Robertson-Walker metrics,

Spatially flat Universe

Friedmann equations (which are Einstein

equations for the FRW metrics)

Scalar and tensor perturbations

Generally, the metrics perturbations can be split

into irreducible representations which correspond

to scalar, vector and tensor perturbations. Sca

lar perturbations describe density

perturbations. Vector perturbations correspond to

perturbations of vortical velocity . Tensor

perturbations correspond to gravitational

waves. Here we focus on scalar perturbations.

Scalar perturbations of the metrics and

energy-momentum tensor.

8 scalar potentials.

Gauge transformations

Splitting in background and perturbation is

not unique. With coordinate transformations, we

obtain different background and different

perturbation. Hence, unphysical perturbations may

arise.

In a bit different reference frame

Gauge-invariant variables

Almost all of the metrics and material potentials

are not gauge- invariant, but one can construct

gauge-invariant variables from them. One of the

variables is q-scalar

(V.N. Lukash, 1980)

q-scalar is constructed from the gravitational

part A which is prominent at large scales and a

hydrodynamical part (second term) which is

prominent at small scales.

Inverse transformations from q-scalar to the

initial potentials

The material potentials and the metrics

potentials are not independent. They are linked

through perturbed Einstein equations

The inverse transformations are as follows

Thus, there are 10 unknowns for 6 equations. We

then set E0 (isotropic pressure), and a

gauge-transformation contains two arbitrary

scalar functions

Now there is the only unknown left.

One extra equation can be obtained in two

ways. The first way (hydrodynamical approach) is

to write a relation between comoving

gauge-invariant perturbations of pressure and

energy density.

The second way (field approach) is to write a

quite arbitrary Lagrangian for the phi-field

Field approach

One immediately has the energy-momentum tensor.

Field approach

Thus, the two approaches are equivalent to first

order.

Dynamical equation for q

With both approaches, we obtain the following

equation for evolution of the q field

In the field approach one should substitute beta

for cs

Action and Lagrangian of perturbations

The perturbation action is quite simple

That is, q is a test massless scalar field.

Isocurvature perturbations

With several media, perturbations that do not

perturb curvature are also possible. These are

isocurvature (isothermic, entropy) perturbations.

The Lagrangian for isocurvature perturbations

Equations of motion

Normal modes

Conclusions

- Hydrodynamical and field approaches are

equivalent to first order of the cosmological

scalar perturbations theory. - The Lagrangian for adiabatic and isocurvature

modes has been built. It appears that the

isocurvature mode also has a speed of sound.

Thank you!